5. TRANSFER FUNCTION FOR MATTER PERTURBATIONS

We now have all the ingredients to evolve the matter perturbation from
an initial value =
i at y
= yi << 1 to the current epoch
y = y0 = aeq-1 in
the matter dominated phase at y >>
1. Initially, the wavelength of the perturbation will be bigger than
the Hubble radius and the perturbation will essentially remain frozen.
When it enters the Hubble radius in the radiation dominated phase, it
begins to grow but only logarithmically (see
section 4.2 ) until the universe becomes
matter dominated. In
the final matter dominated phase, the perturbation grows linearly with
expansion factor. The relation between
final and initial perturbation can be obtained by combining these results.

Usually, one is more interested in the power spectrum
Pk(t) and the
power per logarithmic band in k-space
k(t).
These quantities are defined in terms of
k(t)
through the equations:

(100)

It is therefore convenient to study the evolution of
k3/2k
since its square will immediately give the power per logarithmic band
k2 in k-space.

Let us first consider a mode
which enters the Hubble radius in the radiation dominated phase at the
epoch aenter. From the scaling relation,
aent / ktentaent2 we find that yent
= (keq / k). Hence

(101)

where two factors - as indicated - gives the growth in radiation
(RD) and matter dominated (MD) phases. Let us next consider the modes
that enter in the matter dominated phase. In this case,
aent / ktentaent3/2 so that
yent = (keq /
k)2. Hence

(102)

To proceed further, we need to know the k-dependence of the
perturbation when it enters
the Hubble radius which, of course, is related to the mechanism that
generates the initial power spectrum.
The most natural choice will be that all the modes enter the Hubble
radius with a constant amplitude
at the time of entry. This would imply that the physical perturbations
are scale invariant
at the time of entering the Hubble radius, a possibility that was
suggested by Zeldovich and Harrison
[18]
(years before inflation was invented!). We will see later that this is
also true for perturbations generated by inflation and thus is a reasonable
assumption at least in such models. Hence we shall assume

(103)

Using this we find that the current value of perturbation is given by

(104)

The corresponding power per logarithmic band is

(105)

The form for P(k) shows that the evolution imprints the
scale keq on the power spectrum
even though the initial power spectrum is scale invariant. For
k < keq (for large spatial
scales), the primordial form of the spectrum is preserved and the
evolution only increases the amplitude preserving the shape. For
k > keq (for small spatial scales), the
shape is distorted and in
general the power is suppressed in comparison with larger spatial
scales. This arises because modes with small wavelengths enter the
Hubble radius early on and have to wait till the universe becomes matter
dominated in order to grow in amplitude. This is in contrast to modes
with large wavelengths which continue to grow. It is this effect which
suppresses the power at small wavelengths (for k >
keq) relative to power at larger wavelengths.